If by 'sword' you mean any one-handed weapon, I would suggest you use daggers instead.

If you mean a weapon under the 'sword' category, the nightingale blade is an easy one to obtain. Otherwise, go with the best sword you can get (Daedric, Ebony, Glass, Orcish, Elven, Dwarven, Steel or Iron. In descending order)

If by, "Assassin's Sword" you mean, "Daggers" then the Razor of Mehrunes Dagon can easily be considered one of the best Daggers for an Assassin Build. Reaching Level 15 will provoke a Courier to hand you an invitation to the Museum in Dawnstar, which activates the Daedric Quest.

The "Blade of Woe" however, is usually consider the best Dagger for an Assassin, although you must finish the entire Dark Brotherhood Storyline before obtaining it.

Mehrune's Razor can't be gotten at Level 15, as the quest doesn't start until you hit Level 20.

In terms of Daggers, your best ones are:

Blade of Woe: Tied with the Dragonbone Dagger for best damage. Get 70 Pickpocket and spend 4 perks, and you can get it from Astrid the moment you meet her. Or pickpocket it off of her in the Dark Brotherhood Sanctuary. Going through the Dark Brotherhood is not required.

Dragonbone Dagger: Only available if you have Dawnguard. Same damage as the Blade of Woe, but you can add your own enchantments for even more damage.

Mehrune's Razor: Same damage as a Daedric Dagger, but honestly looks a heck of a lot better. Also, it's chance to kill is supposed to be 1%, but it's actually 2/101, which works out to 1.9%. Also, it's not an enchantment, so you never run out of charge for it.

Swords are generally not a weapon used by Assassins because they're not silent. All weapons except daggers create noise when you swing them, even if you don't hit anything. This can draw the attention of enemies, ruining your stealth and putting you into a fight against a bunch of enemies. However, if you're looking for the best Sword anyway, your best ones are either Chillrend when gotten after Level 46, or a Dragonbone Sword if you have Dawnguard.

Mehrune's Razor can't be gotten at Level 15, as the quest doesn't start until you hit Level 20.

Seriously? Odd. I thought you were able to obtain it at Level 15 for some odd reason. Thanks for the correction on that!

quote swk3000

Blade of Woe: Tied with the Dragonbone Dagger for best damage. Get 70 Pickpocket and spend 4 perks, and you can get it from Astrid the moment you meet her. Or pickpocket it off of her in the Dark Brotherhood Sanctuary. Going through the Dark Brotherhood is not required.

Alternatively, if you simply choose to kill her and Destroy the Dark Brotherhood instead, you can obtain the blade with that method as well.

There's another dagger that's pretty easy to get at very low levels. Just go to Falkreath and ask the Innkeeper about work. If you get a quest to check on some local hunters, you can clear out a pretty easy dungeon and get a steel dagger that has a huge critical hit chance, which is great for stealth kills.